Is it possible to simulate hardware of MIPS architecture computer defined using Verilog/VHDL

Can you clarify what HDL you want to use? The choices are basically Verilog or VHDL, [EDIT], and their relatives, Verilog-ASM and VHDL-ASM (Analog mixed-signal). [/EDIT]Verilog has some C-like syntax, which makes it easier to pick up if you've worked with C before, but this also makes it easy to develop bad habits - You can't program hardware in C, because it's all parallel! Also like C, it assumes you know what you're doing, and it's easy to shoot yourself in the foot. VHDL forces you to think in a totally different way, which is helpful, but difficult. It is more verbose, and more likely to warn you if you do something strange. See this Slashdot discussion, or this article.

EDIT: The "Netlist languages" are not something I've used for design work (in a text editor), but I suppose that you could. SPICE, Cadsoft Eagle's format, and EDIF are all examples (with very different purposes) that come to mind. I've only used netlists to verify that my schematic is correct (does each connection in my Eagle schematic make sense), to tweak the abstraction provided by a simulator (SPICE, similar to the way one uses ASM statements in C), or to do export/import between different programs (EDIF).
The Spectre netlisting language is related to Verilog-A[nalog] and SPICE, and is designed for design and verification work. MAST is a component modeling language which is compatible with Verilog-AMS and VHDL-AMS. Searching for tutorials on these languages shows that tools which look like schematic capture utilities are often used, rather than programming in the netlisting language itself.

I'll also second the Xilinx Spartan FPGA, and a Digilent dev board. However, I'd go with the Basys ($60) or Nexys ($100) if you don't need the Ethernet on the Starter board ($150) mentioned by O Engenheiro (Prices with education discount). The Basys and Nexys are cheaper and therefore more popular in schools, so there are more tutorials and labs online.

Given the goals of the class, I think the TTL approach is fine, and I say this as an "FPGA guy". FPGAs are a sea of logic and you can do all sorts of fun stuff with them, but there's only so much that's humanly possible to do in a semester.

Looking at your syllabus, your class is a mix of the logic design and "machine structures" courses I took in undergrad. (Plus, it's for CS majors. I'm all for CS majors having to face real hardware--letting them get away with writing code seems like a step back.) At this introductory level, where you're going over how assembly instructions are broken down, I see no real benefit to having students do things in code versus by hand. Doing HDL means learning the HDL, learning how to write synthesizable HDL, and learning the IDE. This is a lot more conceptual complexity and re-abstraction. Plus you have to deal with software issues.

Generally the point of a course that uses FPGAs is to practice creating logic that is useful--useful for talking to peripherals, serial comms, RAM, video generators, etc. This is valuable knowledge to have, but it seems very much out of the scope of your course. More advanced classes in computer architecture have students implement sophisticated CPUs in FPGAs, but again, this seems out of the scope of your course.

I would at the very least devote a lecture to FPGAs. Run through a few demos with a dev board and show them the workflow. Since you're at Mills, perhaps you could contact the folks at Berkeley who run CS150/152 and go see how they do things.